#6914549

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907945
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,8)(5,7)(17,18)(19,28)(20,32)(21,27)(22,26)(23,37)(24,30)(25,34)(29,44)(31,40)(33,41)(35,39)(36,42)(38,45)(43,47)(46,48);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,46)(34,47)(36,45)(37,39)(41,44)(42,43);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,34,22,33,21,36)(18,25,27,32,26,42,37,41)(19,29,31,47,30,46,35,45)(24,38,40,48,28,43,39,44);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(18,26)(19,30)(20,33)(27,37)(31,35)(34,36)(38,43)(44,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,26,37)(19,31,30,35)(20,34,33,36)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6821745);